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Home , Neutral mutation, Phenotypic trait

Examining the Impact of Neutral Theory on Genetic Algorithm Population

Seamus Hill and Colm O’Riordan College of Engineering & Informatics, National University of Ireland Galway, Ireland

Keywords: Neutral Theory, , Neutrality, , , Genetic Algorithms.

Abstract: This paper examines the introduction of neutrality as proposed by Kimura (Kimura, 1968) into the genotype- phenotype mapping of a Genetic Algorithm (GA). The paper looks at the evolution of both a simple GA (SGA) and a multi-layered GA (MGA) incorporating a layered genotype-phenotype mapping based on the biological concepts of and . Previous research in comparing GAs often use performance statis- tics; in this paper an analysis of population dynamics is used for comparison. Results illustrate that the MGA population’s evolution trajectory is quite different to that of the SGA population over dynamic landscapes and that the introduction of neutrality implicitly maintains genetic diversity within the population primarily through genetic drift in association with selection.

1 INTRODUCTION ated with a SGA and a multi-layeredGA (MGA). The motivation is to develop a tunable, synonymous, non- Neutral theory as proposed by Kimura (Kimura, trivial GA representation which incorporates neutral- 1968), offered an alternative to the Darwinian view; ity, in order to gain an understanding of the effects of stating that the involved in the evolutionary neutrality on population dynamics. The contributions process are neither advantageous nor disadvantageous are as follows: an analysis of the impact of neutrality to the survival of an individual and that most muta- on population evolution; an examination of the im- tions are caused not by selection, but rather by random pact of neutrality on population variation; and finally genetic drift. However, in (Kimura, 1983), Kimura an illustration of the impact of neutrality on pheno- pointed out that although does play a typic variability. role in adaptive evolution, only a tiny fraction of DNA The paper is laid out as follows: Section 2 gives a changes are adaptive. The vast bulk of mutations are brief backgroundto Neutral theory and the use of neu- phenotypically silent. trality in GAs. Section 3 outlines the MGA used in By adopting the principal of Darwinism, simple the paper, while Section 4 describes the experiments genetic algorithms (SGA), can be viewed as imple- undertaken. Section 5 outlines and analyses the re- menting the process of evolution without containing sults, Section 6 concludes and Section 7 outlines fu- any explicit neutral mutations. In other words, each ture work. is either an advantage or a disadvantage to the individual in terms of fitness, with selection then propagating the fitter individuals. As the search 2 BACKGROUND progresses, the exploration and exploitation ratio de- creases as the population converges. If we are to im- Neutral Theory as discussed by Kimura (Kimura, plement a genetic algorithm (GA) based on the prin- 1968), argues that mutation, not selection, is the main ciples of neutral theory then neutrality needs to be force in evolution. He describes how a mutation from introduced. Neutrality can be viewed as a situation one to another can be viewed as being neutral where a number of different can represent if it does not affect the phenotype, as the number of the same phenotype. Traditionally, GAs are evaluated different genotypes which store genetic information and compared in relation to performance measures. is far greater than the number of . This In this paper, in addition to considering performance, implies that the representation from genotype to phe- the authors examine the population dynamics associ- notype must incorporate an element of redundancy

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Hill, S. and O’Riordan, C.. Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution. In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 1: ECTA, pages 196-203 ISBN: 978-989-758-157-1 Copyright c 2015 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution

and neutral mutations are possible. Mutations can be seen as relating to a collection. Variability, on be viewed as neutral if they change the genotype but the other hand, can be described as the leaning to don’t impact on the phenotype. vary and the “variability of a phenotypic trait de- Generally, neutrality can be viewed as a situation scribes the way in which it changes in response to where the size of the search space is increased, with- environmental and genetic influences” (Wagner and out an equivalent increase in the solution space. This Altenberg, 1996). When exploring the phenotypic results in a situation where a mutated individual, at space, it is critical to gain an understanding of the the genotypic level, can still represent the same phe- variational topology in trying to determine the shape notype. Neutrality should be beneficial when neutral- of the landscape (Toussaint, 2003b). Many evolu- ity changes the search bias in order to improve the tionary algorithms are created using a fixed variation probability of locating the global optimum. topology, in other words you don’t need to track the Although originally viewed as being anti- neutrality to explain the evolution trajectory. How- Darwinian, Kimura (Kimura, 1983) stated that al- ever, in nature, phenotypic variation landscapes are though natural selection is important in evolution, the not fixed. These non-fixed phenotypic variation land- number of DNA changes which are adapted in evo- scapes can be referred to as non-trivial in terms of lution are small, with the vast majority of mutations their genotype-phenotype map (Toussaint, 2003b). A being phenotypically silent. Following Kimura, work non-trivial genotype-phenotype map can be viewed by King and Dukes (King and Dukes, 1969) describes as having the following characteristics: firstly, a phe- how much of the evolution of is down to neu- notype can be encoded by many genotypes and sec- tral mutations and genetic drift. A number of studies ondly, the phenotypic variability of a number of phe- focused on neutral theory (Schuster et al., 1994; Huy- notypes will depend on their genotypes (Toussaint, nen et al., 1996; Huynen, 1996; Schuster, 1997) etc. 2003a). and illustrated that by introducing redundant repre- sentation and thus, neutral mutation, the connectivity The primary inspiration for the multi-layered GA in fitness landscapes can be altered. In other words, can be found in the biological processes of transcrip- when a number of genotypes represent the same phe- tion and translation. At a very basic level, the bio- notype, they can be viewed as a neutral set and in turn logical process of transcription involves the copying alter the way in which a population explore the search of information stored in DNA into an RNA molecule, space. which is complementary to one strand of the DNA. Previous work on GA genotype-phenotype map- The process of translation then converts the RNA, us- pings which introduce neutrality, has produced results ing a predefined translation table, to manufacture pro- indicating the neutrality may prove useful in chang- teins by joining amino acids. These proteins can be ing environments and over more difficult landscapes. viewed as a manifestation of the con- Ebner et al. (Ebner et al., 2001) outlined how high tained within DNA and act as organic catalysts in levels of mutation could be sustained by having neu- anatomy. The MGA includes a layered genotype- tral networks present. They also identified that neutral phenotype map which adopts a basic interpretation networks assist in maintaining diversity in the popula- of the transcription and translation processes and al- tion, which may be advantageous in a changing envi- lows for the implementation of a operator. The genotype search space is represented ronment. Similar findings were found in (Grefenstette l by Φg where Φg = 0,1 and l is the genotype and Cobb, 1993). Toussaint and Igel (Toussaint and { } Igel, 2002) argued that approaches to self-adaption in length. The transcription phase of the MGA maps Φ evolutionary algorithms can be viewed as an example the binary genotype to the DNA search space d, Φ l/2 of the benefits of neutrality, because with non-trivial where d = A,C,G,T , with the following map- pings: 00 {A; 01 C}; 10 G and 11 T . Fol- neutrality, different genotypes which are part of the → → → → same neutral set may have different phenotypic distri- lowing this, a bijective mapping takes place, map- ping the DNA space to an RNA space Φr, where butions. l/2 Φr = A,C,G,U . U is included for biological plausibility{ and has} no impact on the evolution un- less we include operators at this level. Following tran- 3 MULTI-LAYERED GENETIC scription, the translation phase takes place, mapping ALGORITHM (MGA) Φr Φp, where Φp represents the phenotype space → l/c and Φp = 0,1 , where c is the cardinality chosen The concepts of Variation and Variability need to be at initialisation{ } to create a translation table. The level differentiated. Variation can be described as the dif- of redundancy is determined by c and in this paper ference between individuals in a population and can c = 6 (see Figure 1), and implies Φg > Φp where | | | |

197 ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

c > 1. Missense mutationin natureis carriedout at the population will not enable the GA to locate the new RNA level. In relation to the MGA, the Missense mu- optimum. Hamming difference is used as a measure tation mapping is as follows: A U, C G, G A of diversity both for the genotypic and phenotypic and U C. To summarise the→ variation→ operators,→ diversity. In order to examine the population evo- one-point→ crossover and single- occur lution for both the SGA and the MGA, experiments at the genotype level prior to transcription and mis- were conducted over a 4-bit fully deceptive landscape sense mutation takes place before translation. (Whitley, 1991b) which is reversed after generation The MGA introduces a tunable multi-layered GP- 50, allowing the local optimum to become the global map, which allows a haploid GA to exhibit, some of optimum. the characteristics normally associated with a diploid. 4-bit Deceptive Landscape That is a mechanism for allowing or combina- tions of alleles which proved useful in previous gen- 30 25 30 20 erations (Goldberg and Smith, 1987) and thus, main- 25 20 15 10 taining a form of long term memory without the need 15 10 5 to develop a scheme. The MGA popu- 5 0 0 lation consists of a population of haploid individu- 2 1 -2 0 als, which allows for the use of traditional crossover -1 Y 0 -1 X 1 and mutation variation operators on the genotype. 2-2 This differs from the approach used by diploid GAs Figure 2: 4-bit Deceptive Landscape. (DGAs) i.e. (Yang, 2006), where each individual has two and crossover is divided into Figure 2 graphically illustrates the landscape of the two steps and mutation is viewed as being neutral. 4-bit deceptive problem (Whitley, 1991b), with the x Another difference between the MGA mapping and and y co-ordinates indicating the location on the grid. that of a DGA, is that in the DGA, a phenotype al- To analyse the adaptive qualities of both GAs, the lele is made up from a single genotype which landscape reverses after generation 50, Figure 3, illus- is expressed. In the MGA, a single phenotype al- trates the landscape after the change, where the global lele is made from the cardinality incorporated in the optimum becomes the local optimum and visa-versa. genotype, in this paper we use 6-bits. Although the Reversed 4-bit Deceptive Landscape MGA’s GP-map is non-deterministic, the approach differs from that of real-coded binary representation, 30 25 30 which incorporate a gene-strength adjustment mech- 20 25 20 15 anism (Kubalik, 2005). Real-coded binary represen- 10 Fitness 15 10 5 tations can use standard crossover operators, but mu- 5 0 0 tation is implicit due to the gene-strength adjustment 2 1 mechanism (Kubalik, 2005). -2 0 -1 Y 0 -1 X 1 2-2 Figure 3: Reversed 4-bit Deceptive Landscape.

5 RESULTS

The analysis that follows is broken into three parts: the impact of neutrality on population evolution; the impact of neutrality on variation and the impact of Figure 1: 6-bit MGA Representation Mapping. neutrality on phenotypic variability. However, the re- sults begin by comparing both GAs in a conventional manner based on performance. Figure 4, illustrates 4 EXPERIMENTATION the off-line (averaged best fitness) and on-line (av- eraged fitness) performance for both the SGA and Deception is often used in testing GAs and im- the MGA. The results indicate that the changing 4- plies that the search strategy can be misled (Whitley, bit deceptive landscape initially proved equally easy 1991a). As noted in (Morrison and DeJong, 2002), for both the SGA and the MGA. However, after the diversity is critical for GAs, particularly when the landscape changes, the SGA becomes trapped on the landscape is evolving as recombining a homogeneous local optimum, while the MGA succeeds in locating

198 Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution

the global optimum. A Wilcoxon Rank sum test in- 4-bit Deceptive Landscape MGA Population Generation 0 dicates that the results shown in Figure 4, that is both the off-line and on-line comparison between the SGA 30 25 30 20 and MGA, were statistically significant. The remain- 25 20 15 10 ing analysis examines the evolution of both popula- Fitness 15 10 5 tions, in an attempt to gain a better understanding of 5 0 0 the impact on neutrality on population dynamics over 2 1 0 the course of an evolutionary time period and to illus- -2 -1 -1 Y 0 trate the performance of both population in a dynamic X 1 -2 2 environment. Figure 6: MGA - Generation 0. SGA & MGA 4-bit Deceptive Problem - Online/Offline Performance Analysis 35

4-bit Deceptive Landscape SGA Population Generation 50

30

30 25 25 30 20 25 20 15 10 20 Fitness 15 10 5 5 0

Fitness 0 15 3 2 1 -2 0 10 -1 Y 0 -1 1 X 2 -2 3

5 MGA Off-line Performance MGA On-line Performance Figure 7: SGA - Fitness Landscape Generation 50. SGA Off-line Performance SGA On-line Performance 0 20 40 60 80 100 120 140 160 180 200 4-bit Deceptive Landscape MGA Population Generation 50 Generations Figure 4: SGA & MGA On-line/Off-line Performance. 30 25 30 20 25 20 15 10 5.1 Neutrality & Population Evolution Fitness 15 10 5 5 0 0 5.1.1 Analysis Before the Landscape Change 3 2 1 -2 0 -1 Y 0 -1 X 1 -2 In relation to the evolution of the SGA’s population, 2 Figure 5 gives an overview of the population distri- Figure 8: MGA - Fitness Landscape Generation 50. bution over the landscape at generation 0, with the initial population of 20 individuals randomly spread the global optimum. However, due to the genotype- over the landscape. Figure 6 shows the MGA popu- phenotype mapping, the population doesn’t converge lation distribution over the problem landscape. The as much as the SGA. Figure 8, shows the many-to-one initial randomly generated population distribution for representation present in the population, with differ- the MGA is quite similar to that of the SGA. ent genotypes having the same fitness represented by

4-bit Reversed Deceptive Landscape SGA Population Generation 0 different colours and shapes. The MGA population’s evolutionary trajectory

30 differs considerably by not converging on the global 25 30 20 optimum. Therefore the populationconsists of a num- 25 20 15 ber of neutral networks, which is a result of gene flow 10

Fitness 15 10 5 due to the presence of neutrality. 5 0 0

3 2 5.1.2 Analysis After the Landscape Change 1 -2 0 -1 Y 0 -1 X 1 -2 2 Figure 9, illustrates the SGA population distribution Figure 5: SGA - Fitness Landscape Generation 0. when the landscape changes at generation 51 and shows the population, which now converges on the Examining the populations for both GAs at gen- local optimum as the landscape has reversed in rela- eration 50, which is the last generation before the tion to the fitness . It also indicates all but landscape changes, we see that the SGA’s popula- one member of the population are located on the local tion has converged, apart from the impact of mutation optimum. (Figure 7). The MGA’s population has also located The MGA population can be seen distributed on

199 ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

4-bit Reversed Deceptive Landscape SGA Population Generation 200 4-bit Reversed Deceptive Landscape SGA Population Generation 51

30 30 25 30 25 20 30 25 20 25 20 15 15 10 20 Fitness 15 10 10 5

Fitness 15 10 5 5 0 0 5 0 0 3 2 3 -2 1 2 -1 0 Y 0 -2 1 1 -1 X 2 -1 0 Y -2 0 3 1 -1 X 2 3-2 Figure 11: SGA - Fitness Landscape Generation 200. Figure 9: SGA - Fitness Landscape Generation 51. 4-bit Reversed Deceptive Landscape MGA Population Generation 200

4-bit Reversed Deceptive Landscape MGA Population Generation 51

30 25 30 30 20 25 25 30 20 15 20 25 10

Fitness 15 15 20 10 5 10 Fitness 15 5 0 10 5 0

5 0 2 0 1 3 0 -2 2 -1 -1 Y 1 0 1 -2 -2 0 X -1 Y 2 0 -1 1 X 2 -2 3 Figure 12: MGA - Fitness Landscape Generation 200. Figure 10: MGA - Fitness Landscape Generation 51.

SGA Generation 0 - Population Hamming Distance the problem landscape in Figure 10, with the bulk of the population located on the local optimum. The fig- 1 1 0.8 ure also indicates that due to the presence of neutrality 0.8 0.6 in the representation, the population is dispersed over 0.6 0.4 0.4 0.2 a wider number of fitness plateaus. Hamming Distance 0.2 0 0

Examining the population evolution through to 19 20 17 18 15 16 13 14 generation 200 for both GAs. The SGA population 0 11 12 1 2 3 9 10 4 5 6 8 Genotypes/Phenotypes 7 8 6 7 9 10 11 5 12 13 14 3 4 remains trapped on the local optimum (see Figure 11), Genotypes/Phenotypes 15 16 2 17 18 19 0 1 while the MGA population remains clustered around 20 the global optimum, see Figure 12. This appears to in- Figure 13: SGA Genotypic/Phenotypic Diversity Genera- dicate that the MGA, through the genotype-phenotype tion 0. mapping, implicitly maintains a level of genetic di- MGA Generation 0 - Genotype Population Hamming Distance versity within the population and is resistant to con- vergence, thereby offering the ability to adapt in a dy- 1 namic environment. The impact of neutrality on pop- 1 0.8 ulation evolution is described in Sub-section 5.1 and 0.8 0.6 0.6 0.4 illustrated the population distribution for the SGA and 0.4 0.2 Hamming Distance 0.2 0 MGA, both before and after the landscape changed. 0 20 18 19 16 17 14 15 12 13 0 1 10 11 2 3 4 9 5 6 7 8 Genotypes 7 8 9 6 5.2 Neutrality & Variation 10 11 4 5 12 13 14 3 Genotypes 15 16 17 1 2 18 19 20 0 As a traditional SGA maps directly from the genotype Figure 14: MGA Genotypic Diversity - Generation 0. to the phenotype, both the genotypic search space and the phenotypic search space are identical. With of diversity present in both populations. This can the MGA mapping, neutrality increases the geno- be explained as evolution has not yet begun and the typic search space in comparison to the phenotypic populations have been randomly generated. Also, as search space. Using normalised Hamming distances neutrality is introduced the MGA’s genotypic search between individuals, Figure 13 and Figure 14, il- space increases, while the MGA’s phenotypic space lustrates the population diversity for both the SGA (Figure 15) is quite similar to that of the SGA. search spaces and the MGA genotypic search spaces, As the populations evolve to generation 50, con- respectively, at generation 0. From the figures, it ap- vergence has occurred in the SGA (apart from the in- pears that at generation 0, there is quite a large level fluence of mutation) and the population is shown in

200 Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution

MGA Generation 0 - Phenotype Population Hamming Distance MGA Generation 50 - Phenotype Population Hamming Distance

1 1

1 0.8 1 0.8

0.8 0.6 0.8 0.6

0.6 0.4 0.6 0.4

0.4 0.2 0.4 0.2

Hamming Distance 0.2 0 Hamming Distance 0.2 0 0 0

19 20 19 20 17 18 17 18 15 16 15 16 13 14 13 14 0 11 12 0 11 12 1 2 3 9 10 1 2 3 9 10 4 5 6 8 Phenotypes 4 5 6 8 Phenotypes 7 8 6 7 7 8 6 7 9 10 11 5 9 10 11 5 12 13 14 3 4 12 13 14 3 4 Phenotypes 15 16 17 1 2 Phenotypes 15 16 17 1 2 18 19 20 0 18 19 20 0 Figure 15: MGA Phenotypic Diversity - Generation 0. Figure 18: MGA Phenotypic Diversity - Generation 50.

SGA Generation 50 - Population Hamming Distance SGA Generation 200 - Population Hamming Distance

1 1

1 0.8 1 0.8

0.8 0.6 0.8 0.6

0.6 0.4 0.6 0.4

0.4 0.2 0.4 0.2

Hamming Distance 0.2 0 Hamming Distance 0.2 0 0 0

19 20 19 20 17 18 17 18 15 16 15 16 13 14 13 14 0 11 12 0 11 12 1 2 3 9 10 1 2 3 9 10 4 5 6 8 Genotypes/Phenotypes 4 5 6 8 Genotypes/Phenotypes 7 8 6 7 7 8 6 7 9 10 11 5 9 10 11 5 12 13 14 3 4 12 13 14 3 4 Genotypes/Phenotypes 15 16 17 1 2 Genotypes/Phenotypes 15 16 17 1 2 18 19 20 0 18 19 20 0 Figure 16: SGA Genotypic/Phenotypic Diversity Genera- Figure 19: SGA Genotypic/Phenotypic Diversity Genera- tion 50. tion 200.

MGA Generation 50 - Genotype Population Hamming Distance MGA Generation 200 - Genotype Population Hamming Distance

1 1

1 0.8 1 0.8

0.8 0.6 0.8 0.6

0.6 0.4 0.6 0.4

0.4 0.2 0.4 0.2

Hamming Distance 0.2 0 Hamming Distance 0.2 0 0 0

19 20 19 20 17 18 17 18 15 16 15 16 13 14 13 14 0 11 12 0 11 12 1 2 3 9 10 1 2 3 9 10 4 5 6 8 Genotypes 4 5 6 8 Genotypes 7 8 6 7 7 8 6 7 9 10 11 5 9 10 11 5 12 13 14 3 4 12 13 14 3 4 Genotypes 15 16 17 1 2 Genotypes 15 16 17 1 2 18 19 20 0 18 19 20 0 Figure 20: MGA Genotypic Diversity - Generation 200. Figure 17: MGA Genotypic Diversity - Generation 50.

MGA Generation 200 - Phenotype Population Hamming Distance Figure 16. With the MGA population, genetic diver- sity is maintained within the genotypic search space, 1 1 0.8 see Figure 17. The MGA phenotypic search space 0.8 0.6 (Figure 18) illustrates the diversity at a phenotypic 0.6 0.4 0.4 0.2 level. Hamming Distance 0.2 0 At generation 200, the SGA population remains 0 19 20 17 18 15 16 13 14 on the local optimum and contains little diversity to 0 11 12 1 2 3 9 10 4 5 6 8 Phenotypes 7 8 6 7 9 10 11 5 12 13 3 4 allow it to escape. This situation is shown in Fig- 14 15 16 2 Phenotypes 17 18 0 1 ure 19, reflecting the lack of diversity within the pop- 19 20 ulation when viewed as a function of Hamming dis- Figure 21: MGA Phenotypic Diversity - Generation 200. tance. The MGA genotypic population diversity at gen- 5.3 Neutrality & Phenotypic Variability eration 200, is shown in Figure 20 and the pheno- typic diversity is illustrated in Figure 21. The level of The final examination of the effects of neutrality on phenotypic diversity present, relates to the many-to- the evolutionary trajectory of the MGA population re- one genotype-phenotype mapping and to individuals lates to the impact of mutation. In order to examine of the population being part of other neutral networks. the effect of single-bit mutation and missense muta- tion, we randomly selected an individual from both

201 ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

the SGA and MGA populations from various gen- MGA Genotype - 010000011000010000111000 - Fitness 30 - Bit Mutation & Missense Mutation erations. We then flipped all of the bits, one at a time, in sequence and measured the impact on fitness. 30 25 30 20 The aim of this approach is to examine how the pres- 25 20 15 10

Fitness 15 ence of neutrality impacts on mutation in GAs which 10 5 5 0 have synonymous representations. A representation 0 3 is viewed as synonymously redundant if the geno- 2 -2 1 -1 0 Y 0 types representing the same phenotype have the same 1 -1 X 2 properties and are next to one another in the muta- 3 -2 tion space. With the MGA population, a genotype, Figure 24: MGA 1-Point & Missense Mutated Individual. when mutated, can produce either a silent or an adap- tive single-bit mutation and a silent or adaptive mis- MGA Genotype - 100010011110010000001010 - Fitness 30 - Bit Mutation & Missense Mutation sense mutation. 30 25 Figure 22 shows the impactof mutationin an SGA 30 20 25 population. Figure 23 visualises the impact of single- 20 15 10

Fitness 15 bit mutation on an individual from the MGA popu- 10 5 5 0 lation. The phenotypic distribution for both GAs is 0 3 2 similar as the neutrality associated with the MGA rep- -2 1 -1 0 Y 0 1 -1 resentation is synonymous. X 2 3 -2 Genotype 2 - 1111 - One Bit Mutation Neighbours - Generation 0 Figure 25: MGA 1-Point & Missense Mutated Individual.

30 show that the MGA population allows individuals re- 25 30 20 25 siding on the same fitness plateau, have different phe- 20 15 10 notypic distributions. This section highlighted the ef-

Fitness 15 10 5 5 0 fect of mutation, with results illustrating that the SGA 0

2 can only access local plateaus, while the MGA, which 1 can be phenotypically silent, has greater variability. -2 0 -1 Y 0 -1 Y 1 2-2 Figure 22: SGA 1-Point Mutated Individual. 6 CONCLUSION

MGA Genotype 3 - 001100101000110110001000 - One Bit Mutation Neighbours - Generation 50 A population’s ability to survive in dynamic environ-

30 mentsoften dependson a level of diversityto be main- 25 30 20 tained within the population. As a GA search involves 25 20 15 a mapping between the genotype and the phenotype, 10

Fitness 15 10 5 a SGA, through it’s one-to-one genotype-phenotype 5 0 0 mapping, quickly eliminates diversity from the popu- 3 2 lation through its selection policy and low mutation -2 1 -1 0 Y 0 1 -1 rates. The results presented, illustrate that through X 2 3-2 the implementation of Neutral theory, as proposed Figure 23: MGA 1-Point Mutated Individual. by Kimura (Kimura, 1968), the genotype-phenotype mapping of the MGA allows for a tunable, non-trivial, The MGA representation includes missense muta- many-to-one relationship. The contribution of this tion which operates within the layers of the genotype- form of mapping is the implicit maintenance of re- phenotype mapping and allows the phenotypic vari- lated genetic diversity within the population, which ability to differ for individuals having the same level allows the occupation by the population, of a greater of fitness. Figure 24 illustrates the phenotypic distri- number of fitness plateaus. By adopting this approach bution for individual 010000011000010000111000, convergence at a phenotypic level can be achieved, showing that the phenotypic distribution for the MGA but genetic diversity is maintained at a genotypic is greater than that of the SGA, as individuals are level. Neutral theory (Kimura, 1968), would suggest occupying more fitness plateaus. Figure 25, il- that where genetic changes spread across a popula- lustrates the phenotypic distribution for individual tion, changes may or may not alter the phenotype and 100010011110010000001010, which resides on the are a result of genetic drift. The results indicated same neutral network with a fitness level of 30. They that neutrality, as introduced by the MGA mapping,

202 Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution

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